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In practice with investments, analysts frequently need to find present values indexed at times other than *t* = 0. Subscripting the present value and evaluating a perpetuity beginning with $100 payments in Year 2, we find PV_{1} = $100/0.05 = $2,000 at a 5 percent discount rate. Further, we can calculate today’s PV as PV_{0} = $2,000/1.05 = $1,904.76.

Consider a similar situation in which cash flows of $6 per year begin at the end of the 4th year and continue at the end of each year thereafter, with the last cash flow at the end of the 10th year. From the perspective of the end of the third year, we are facing a typical seven-year ordinary annuity. We can find the present value of the annuity from the perspective of the end of the third year and then discount that present value back to the present. At an interest rate of 5 percent, the cash flows of $6 per year starting at the end of the fourth year will be worth $34.72 at the end of the third year (*t* = 3) and $29.99 today (*t* = 0).

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